Hi!
I know that the term 'reduced (nonfinite) clauses' is frequently used, although some prefer the more traditional label 'participle phrase'. But the problem is the other. I questioned whether it is correct and possible to call the phrase (or the clause) the reduced one if this phrase(clause) is gotten not by taking away the pronoun and the auxiliary verb from the full clause but the other way.
Compare,"
The man who was standing near the car was holding the newspaper"(no doubt,"standing near the car" would be really the reduced related clause")
But if you change the sentence"The people who win the game will receive this prize" for "The people winning the game will receive this prize" you have the participle phrase "winning the game" which is not the part of the full clause, that is, it is gotten not by taking away two words.
Yet another example,
Because he didn't have free time he couldn't say "goodbye".
Having no free time he couldn't say "goodbye".
Again, "having no free time" is not the reduced part of "because he didn't have free time". It is not the part of it at all. So, I wonder if one could call "having no free time" and "winning the game" the reduced clauses. I hope you get me.
Thanks in advance.

The term 'reduced clause', probably for the sake of simplicity, tends to be used fairly indiscriminately in EFL textbooks to refer both to clauses that can be considered truly elliptical (as your first example) and to those where an actual change of wording - as opposed to simple insertion of omitted words - is required to maintain equivalence (as your second).

This may indeed be sacrificing accuracy for simplicity, and, for that reason, some grammarians (myself included!) prefer to avoid the term, treating the two alternative constructions (the participial, on the one hand, and the relative clause, on the other) as essentially syntactically distinct categories of modifier.